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Properties

Label 18.8.11893977448...5424.2
Degree $18$
Signature $[8, 5]$
Discriminant $-\,2^{6}\cdot 3^{18}\cdot 7^{14}\cdot 29^{4}$
Root discriminant $36.29$
Ramified primes $2, 3, 7, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T766

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![512, -2304, 3072, 576, -5088, 4896, -1736, -384, 948, -824, 474, -96, -217, 306, -159, 9, 24, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 24*x^16 + 9*x^15 - 159*x^14 + 306*x^13 - 217*x^12 - 96*x^11 + 474*x^10 - 824*x^9 + 948*x^8 - 384*x^7 - 1736*x^6 + 4896*x^5 - 5088*x^4 + 576*x^3 + 3072*x^2 - 2304*x + 512)
 
gp: K = bnfinit(x^18 - 9*x^17 + 24*x^16 + 9*x^15 - 159*x^14 + 306*x^13 - 217*x^12 - 96*x^11 + 474*x^10 - 824*x^9 + 948*x^8 - 384*x^7 - 1736*x^6 + 4896*x^5 - 5088*x^4 + 576*x^3 + 3072*x^2 - 2304*x + 512, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 24 x^{16} + 9 x^{15} - 159 x^{14} + 306 x^{13} - 217 x^{12} - 96 x^{11} + 474 x^{10} - 824 x^{9} + 948 x^{8} - 384 x^{7} - 1736 x^{6} + 4896 x^{5} - 5088 x^{4} + 576 x^{3} + 3072 x^{2} - 2304 x + 512 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-11893977448786775020410575424=-\,2^{6}\cdot 3^{18}\cdot 7^{14}\cdot 29^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.29$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{1}{8} a^{8} - \frac{3}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{8} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{1}{8} a^{10} - \frac{1}{16} a^{9} + \frac{1}{16} a^{8} - \frac{1}{4} a^{7} + \frac{1}{16} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{11} - \frac{1}{16} a^{10} + \frac{3}{16} a^{8} + \frac{3}{16} a^{7} + \frac{1}{16} a^{6} - \frac{1}{8} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{32} a^{14} - \frac{1}{32} a^{13} + \frac{1}{32} a^{11} + \frac{1}{32} a^{10} + \frac{1}{16} a^{9} + \frac{7}{32} a^{8} - \frac{1}{2} a^{7} - \frac{7}{16} a^{6} - \frac{1}{4} a^{5} - \frac{3}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{448} a^{15} - \frac{1}{64} a^{14} - \frac{5}{224} a^{13} + \frac{9}{448} a^{12} + \frac{3}{448} a^{11} - \frac{3}{112} a^{10} - \frac{37}{448} a^{9} - \frac{33}{224} a^{8} + \frac{29}{224} a^{7} + \frac{31}{112} a^{6} + \frac{29}{112} a^{5} + \frac{5}{56} a^{4} - \frac{5}{56} a^{3} + \frac{11}{28} a^{2} - \frac{1}{2} a + \frac{1}{7}$, $\frac{1}{45057152} a^{16} - \frac{729}{6436736} a^{15} + \frac{73593}{22528576} a^{14} + \frac{426197}{45057152} a^{13} - \frac{753981}{45057152} a^{12} - \frac{161957}{5632144} a^{11} - \frac{5629521}{45057152} a^{10} - \frac{1764509}{22528576} a^{9} + \frac{2169847}{22528576} a^{8} - \frac{2166805}{11264288} a^{7} - \frac{3316319}{11264288} a^{6} + \frac{2073573}{5632144} a^{5} + \frac{2565285}{5632144} a^{4} - \frac{907651}{2816072} a^{3} + \frac{49237}{201148} a^{2} + \frac{145758}{352009} a + \frac{14368}{50287}$, $\frac{1}{90114304} a^{17} - \frac{1}{90114304} a^{16} + \frac{14943}{22528576} a^{15} + \frac{885981}{90114304} a^{14} + \frac{2109305}{90114304} a^{13} - \frac{776757}{45057152} a^{12} + \frac{697027}{90114304} a^{11} + \frac{172819}{1408036} a^{10} - \frac{88385}{45057152} a^{9} + \frac{2531491}{11264288} a^{8} + \frac{1365055}{22528576} a^{7} + \frac{104301}{2816072} a^{6} + \frac{2253949}{11264288} a^{5} + \frac{749851}{2816072} a^{4} - \frac{332077}{1408036} a^{3} + \frac{97569}{1408036} a^{2} + \frac{157442}{352009} a + \frac{105676}{352009}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $12$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 15152187.4073 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T766:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 144 conjugacy class representatives for t18n766 are not computed
Character table for t18n766 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.13632439166829.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ $18$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.6.6.4$x^{6} + x^{2} + 1$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
3Data not computed
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$